arXiv subjects
Publications and source records attributed to A. Varchenko.
The quantized Knizhnik-Zamolodchikov equation is a difference equation defined in terms of rational $R$ matrices. We describe all singularities of hypergeometric solutions to the qKZ equations.
We consider the quantized Knizhnik-Zamolodchikov-Bernard difference equation (qKZB) with step $p$ and values in a tensor product of finite dimensional evaluation modules over the elliptic quantum group $E_{τ,η}(sl_2)$, the equation defined in terms of elliptic dynamical R-matrices. We solve the equation in terms of multidimensional q-hypergeometric integrals and describe its monodromy properties. We identify the space of solutions of the qKZ equation with the space of functions with values in the tensor product of the corresponding modules over the elliptic quantum group $E_{p,η}(sl_2)$.
We consider the $sl(2)$ quantized Knizhnik-Zamolodchikov equation (qKZ), defined in terms of rational R-matrices. The properties of the equation change when the step of the equation takes a resonance value. In this case the discrete connection defined by the qKZ equation has a invariant subbundle which we call the subbundle of quantized conformal blocks. Solutions of the qKZ equation were constructed in [TV1], [MV1] in terms of multidimensional hypergeometric integrals. In this paper we show that for a resonance step all hypergeometric solutions take values in the subbundle of quantized conformal blocks, moreover the values span the subbundle of quantized conformal blocks under certain conditions. We describe the space of hypergeometric solutions in terms of the quantum group $U_qsl(2)$.
We consider the discrete Knizhnik-Zamolodchikov connection (qKZ) associated to $gl(N)$, defined in terms of rational R-matrices. We prove that under certain resonance conditions, the qKZ connection has a non-trivial invariant subbundle which we call the subbundle of quantized conformal blocks. The subbundle is given explicitly by algebraic equations in terms of the Yangian $Y(gl(N))$ action. The subbundle is a deformation of the subbundle of conformal blocks in CFT. The proof is based on an identity in the algebra with two generators $x,y$ and defining relation $xy=yx+yy$.
We consider the quantized Knizhnik-Zamolodchikov difference equation (qKZ) with values in a tensor product of irreducible sl(2) modules, the equation defined in terms of rational R-matrices. We solve the equation in terms of multidimensional q-hypergeometric integrals. We identify the space of solutions of the qKZ equation with the tensor product of the corresponding modules over the quantum group $U_qsl(2)$. We compute the monodromy of the qKZ equation in terms of the trigonometric R-matrices.
We consider a function $U=e^{-f_0}\prod_j^N f_j^{α_j}$ on a real affine space, here $f_0,..,f_N$ are linear functions, $α_1, ...,α_N$ complex numbers. The zeros of the functions $f_1, ..., f_N$ form an arrangement of hyperplanes in the affine space. We study the period matrix of the hypergeometric integrals associated with the arrangement and the function $U$ and compute its determinant as an alternating product of gamma functions and critical points of the functions $f_0,..., f_N$ with respect to the arrangement. In the simplest example, $N=1, f_0=f_1=t$, the determinant formula takes the form $\int_0^\infty e^{-t} t^{α-1} dt=Γ(α).$ We also give a determinant formula for Selberg type exponential integrals. In this case the arangements of hyperplanes is special and admits a symmetry group, the period matrix is decomposed into blocks corresponding to different representations of the symmetry group on the space of the hypergeometric integrals associated with the arrangement. We compute the determinant of the block corresponding to the trivial representation.
In this note we strenghten a theorem by Esnault-Schechtman-Viehweg which states that one can compute the cohomology of a complement of hyperplanes in a complex affine space with coefficients in a local system using only logarithmic global differential forms, provided certain "Aomoto non-resonance conditions" for monodromies are fulfilled at some "edges" (intersections of hyperplanes). We prove that it is enough to check these conditions on a smaller subset of edges. We show that for certain known one dimensional local systems over configuration spaces of points in a projective line defined by a root system and a finite set of affine weights (these local systems arise in the geometric study of Knizhnik-Zamolodchikov differential equations), the Aomoto resonance conditions at non-diagonal edges coincide with Kac-Kazhdan conditions of reducibility of Verma modules over affine Lie algebras.
We give an integral representation for solutions to the quantized Knizhnik- Zamolodchikov equation (qKZ) associated with the Lie algebra $gl_{N+1}$. Asymptotic solutions to qKZ are constructed. The leading term of an asymptotic solution is the Bethe vector -- an eigenvector of the transfer-matrix of a quantum spin chain model. We show that the norm of the Bethe vector is equal to the product of the Hessian of a suitable function and an explicitly written rational function. This formula is a generalization of the Gaudin-Korepin formula for a norm of the Bethe vector. We show that, generically, the Bethe vectors form a base for the $gl_2$ case.
The Knizhnik-Zamolodchikov equation associated with $s\ell_2$ is considered. The transition functions between asymptotic solutions to the Knizhnik-Zamolodchikov equation are described. A connection between asymptotic solutions and the crystal base in the tensor product of modules over the quantum group $U_qs\ell_2$ is established, in particular, a correspondence between the Bethe vectors of the Gaudin model of an inhomogenious magnetic chain and the $\Bbb Q-$basis of the crystal base.